% !TEX program = xelatex
\documentclass{article}
\usepackage{xeCJK}
\usepackage{amsmath}
\title{习题二T13(1,2),T14(1,2)}
\author{2020141460280张家帅}

\begin{document}
\maketitle
\section{习题二}
\subsection{T13}
\subsubsection{(1)}
\begin{align}
                      & (\exists x)(\exists y)[P(x)\wedge Q(y)]\notag \\
    \Leftrightarrow{} & (\exists x)P(x)\wedge (\exists x)Q(x)\notag   \\
    \Rightarrow{}     & (\exists x)P(x)\notag
\end{align}
\subsubsection{(2)}
\begin{align}
                      & \neg((\exists x)P(x)\wedge Q(a))\notag    \\
    \Leftrightarrow{} & \neg(\exists x)P(x)\vee \neg Q(a)\notag   \\
    \Leftrightarrow{} & (\exists x)P(x)\rightarrow\neg Q(a)\notag \\
\end{align}
\subsection{T14}
\subsubsection{(1)}
\begin{align}
                      & P(x)\wedge (\forall x)Q(x)\notag   \\
    \Leftrightarrow{} & P(y)\wedge (\forall x)Q(x)\notag   \\
    \Leftrightarrow{} & (\forall x)Q(x)\wedge P(y)\notag   \\
    \Leftrightarrow{} & (\forall x)(Q(x)\wedge P(y))\notag \\
    \Rightarrow{}     & (\exists x)(P(x)\wedge Q(x))\notag
\end{align}
\subsubsection{(2)}
\begin{align}
                      & (\exists x)P(x)\rightarrow(\forall x)Q(x)\notag \\
    \Leftrightarrow{} & \neg(\exists x)P(x)\vee(\forall x)Q(x)\notag    \\
    \Leftrightarrow{} & (\forall x)\neg P(x)\vee(\forall x)Q(x)\notag   \\
    \Leftrightarrow{} & (\forall x)(\neg P(x)\vee Q(x))\notag           \\
    \Leftrightarrow{} & (\forall x)(P(x)\rightarrow Q(x))\notag
\end{align}
\end{document}